In the applied sciences one is often confronted with free boundaries, which arise when the solution to a problem consists of a pair: a function u (often satisfying a partial differential equation (PDE)), and a set where this function has a specific behavior. Two central issues in the study of free boundary problems and related problems in calculus of variations and geometric measure theory are:
(1) What is the optimal regularity of the solution u?
(2) How smooth is the free boundary (or how smooth is a certain set related to u)?
In this talk, I will overview recent developments in obstacle type problems and almost minimizers of Bernoulli-type functionals, illustrating techniques that can be used to tackle questions (1) and (2) in various settings.
The study of the classical obstacle problem - one of the most renowned free boundary problems - began in the ’60s with the pioneering works of G. Stampacchia, H. Lewy and J. L. Lions. During the past five decades, it has led to beautiful and deep developments in the calculus of variations and geometric partial differential equations. Nowadays obstacle type problems continue to offer many challenges and their study is as active as ever.
While the classical obstacle problem arises from a minimization problem (as many other PDEs do), minimizing problems with noise lead to the notion of almost minimizers. Interestingly, though deeply connected to "standard" free boundary problems, almost minimizers do not satisfy a PDE as minimizers do, requiring additional tools from geometric measure theory to address (1) and (2).